Solved – Determine paramaters for SARIMA model

acf-pacfarimalagsmodel selectiontime series

I have the following timeseries with a frequency of 12 (months).

Since there is both a trend and seasonality, I differenced the timeseries.
To determine the parameters p, q, P and Q for the SARIMA(p, 1, q)(P, 1, Q)_12 model, I look at the ACF and PACF of the differenced timeseries, shown below.

Now how do I determine the values for p, q, P and Q? I am having trouble reading the ACF and PACF. My guess is parameter P is 0 because the PACF does not show spikes at lag 24 and parameter Q is 2 because the ACF shows 2 spikes after lag 0, 12 and 24. Am I correct so far? About parameters p and q I am clueless.

As a note: auto.arima() gives a SARIMA(1,1,2)(0,1,2)_12 model.

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P is 0? P is 1 because acf shows declining acf at 12,24 36 and pacf shows spike at 12. When modeling seasonal series, it's usually easier to work on either non-seasonal side or seasonal side separately. Not both at same time. Pick which ever is most obvious in acf or pacf. Then proceed step by step. Clear up seasonal side then work on non-seasonal side or vice versa. So, here fit seasonal AR(1), then look at the residual acf and pacf for more clues. Keep adding terms step by step until residuals are white noise. Then you can try overfitting; adding an extra term and check whether it is significant.

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Solved – Time series analysis (ACF, PACF)

It looks like a plausible model, though I'd possibly start with seasonal differencing and see if the residuals from that didn't look over-differenced.

If you do go with the suggested model you have, you'd still look to see if there was autocorrelation in the residuals.

Solved – ACF and PACF of residuals to determine ARIMA model

The thread Terms "cut off" and "tail off" about ACF, PACF functions on this forum will come in handy, Benjamin!

In looking at your plots, I see that the PACF cuts off after 2 lags and the ACF 'decays' towards zero. As per the above thread, that would suggest an AR(2) process for the residuals from your initial regression model. In general, the 'decay' in the ACF can look like what you have in your plot (i.e., exponential decay) or have some sort of sinusoidal flavour, as seen in What does my ACF graph tell me about my data?, for instance.

If you work in R, you could try to fit an ARIMA process to your residuals using the auto.arima function from the forecast package just to see how 'close' your guess (or mine) that the residuals follow an AR(p) process - where p = 4 in your guess or p = 2 in mine - would be to what auto.arima comes with after automated selection of a time series model from the ARIMA class for the model residuals. Just use something like this:

install.packages("forecast")
require(forecast)

auto.arima(residuals(initialreg))

and see what comes out. Another thing to keep in mind is the length of your time series of regression residuals - if that series is not too long, you'll know upfront you won't be able to fit an AR model with too many parameters to it. In particular, you might be able yo fit an AR(1) or AR(2) model, but not an AR(4) or AR(5). The shorter the series, the less complex the AR model that it can support.

Of course, after you fit your AR(2) model to the regression residuals, you have to look at diagnostic plots of the AR(2) model residuals to make sure they look fine (i.e., like white noise).

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Solved – Time series analysis (ACF, PACF)

Solved – ACF and PACF of residuals to determine ARIMA model

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